Wave reflections in a semi-infinitely long cylinder with an attached solid of revolution

Abstract

A numerical procedure is presented for the study of end reflections in a semi-infinitely long isotropic circular cylinder with attached piezoelectric patch. The hybrid method which combines the finite element formulation in the piezoelectric patch with a wave function expansion representation in the isotropic cylinder is employed in the study. The global solution is obtained by imposing the continuity conditions on the displacements and tractions at the interface between the piezoelectric patch and the cylinder. To obtain the wave functions in the cylinder, the governing equations of the cylinder are discretized by a semi-analytical finite element formulation where the discretization occurs through the cylinder's thickness. Solutions in the cylinder are constructed with modal data from a spectral decomposition of the differential equations governing its natural vibrations. These modal data consist of all propagating modes and edge vibrations, constituting the basis for a wave function expansion of the reflection of waves arriving at the end of the cylinder. On the other hand, the piezoelectric patch is discretized by the axisymmetric finite element formulation. Both least-square and virtual work methods are used for evaluating the amplitudes of the reflected wave field. A computer code is developed in the study. Numerical cases are presented to demonstrate the effectiveness and accuracy of the code. The reflections due to monochromatic incoming axisymmetric and flexural wave are studied. For an oscillating end voltage that is out-of-phase with the incoming wave, it is possible to extract electrical energy which is called as energy harvesting. By applying appropriate voltage in the piezoelectric patch, the reflected propagating waves in the cylinder can be eliminated efficiently which is called as passive-control. Cases of such an oscillating voltage with a particular radial distribution are given. Results presented in this study are for different thickness of piezoelectric patch, boundary condition and distribution of applied voltage. The results illustmte the amount of extracted energy as a function of the frequency of the incident monochromatic wave. The study has potential to apply in NDE (nondestructive evaluation), energy harvest and USM (ultrasonic motor).